Local Langlands functoriality for Yu's supercuspidals I: Kaletha's parametrization

This is the first of two papers dedicated to the explicit computation of the Fargues–Scholze correspondence. In this first part, we construct (inspired by, and extending, Kaletha’s explicit Local Langlands Correspondence for non-singular supercuspidal representations) semisimple inertial L-parameters for all cuspidal representations arising from Yu’s constructions, with arbitrary coefficients. We then establish a characterization of this correpondence in terms of certain functoriality properties. This characterization will be used in the sequel paper in order to compute the Fargues–Scholze correspondence after restriction to inertia.